reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem
  for X being RealBanachSpace, f being Function of X,X st
      ex n0 being Element of NAT st iter(f,n0) is contraction holds
   ex xp being Point of X st f.xp = xp &
      for x being Point of X st f.x = x holds xp = x
  proof
    let X be RealBanachSpace;
    let f be Function of X,X;
    assume ex n0 be Element of NAT st iter(f,n0) is contraction;
    then f is with_unique_fixpoint by Th7;
    then consider xp being set such that
A1: xp is_a_fixpoint_of f & for y being set st y is_a_fixpoint_of f holds
      xp = y;
    xp in dom f by A1;
    then reconsider xp as Point of X;
    take xp;
    thus f.xp = xp by A1;
    let x be Point of X;
    assume f.x = x;
    then x is_a_fixpoint_of f;
    hence xp = x by A1;
 end;
